LMIs in Control/Matrix and LMI Properties and Tools/Frobenius Norm

Frobenius Norm
Consider $$A\in \R^{n\times m} $$ and $$\gamma\in \R$$>0

The Frobenius norm of $$A $$ is $$||A|| $$F =$$\sqrt{tr(A^{T}A)}=\sqrt{tr(AA^{T})}$$

The Frobenius norm is less than or equal to $$\gamma$$ if and only if any of the following equivalent conditions are satisfied.

1.There exists $$S\in \R^{n} $$ such that
 * $$ \begin{bmatrix} Z & A^{T} \\

* & 1\\                           \end{bmatrix} \ge0,

$$



\begin{align} \qquad tr(Z)\le\gamma^{\text{2}}. \qquad   \\ \end{align}                              $$

2.There exists $$S\in \R^{m} $$ such that
 * $$ \begin{bmatrix} Z & A \\

* & 1\\                           \end{bmatrix} \ge0,

$$



\begin{align} \qquad tr(Z)\le\gamma^{\text{2}}. \qquad   \\ \end{align}                              $$